The existence and uniqueness of solutions of the Cauchy problem to a stochastic parabolic integro-differential equation is investigated. The equation considered arises in a nonlinear filtering problem with a jump signal process and continuous observation.

We consider a nearest neighbor random walk on the one-dimensional integer lattice with drift towards the origin determined by an asymptotically vanishing function of the number of visits to zero. We show the existence of distinct regimes according to the rate of decay of the drift. In...

We address a constrained utility maximization problem in an incomplete market for a utility function defined on the whole real line. We extend current research in two directions, firstly we allow for constraints on the portfolio process. Secondly we prove our results without relying on the...

We consider the motion of a Brownian particle in , moving between a particle fixed at the origin and another moving deterministically away at slow speed [epsilon]0. The middle particle interacts with its neighbours via a potential of finite range b0, with a unique minimum at a0, where b<2a. We say that the chain of particles breaks on the left- or right-hand side when the middle particle is at a distance greater than b from its left or right neighbour, respectively. We study the asymptotic location of the first break of the chain in the limit of small noise, in the case where [epsilon]=[epsilon]([sigma]) and [sigma]>0 is the...</2a.>

We provide a condition in terms of a supermartingale property for a functional of the Markov process, which implies (a) f-ergodicity of strong Markov processes at a subgeometric rate, and (b) a moderate deviation principle for an integral (bounded) functional. An equivalent condition in terms of...

The aim of the present paper is to study the semimartingale property of continuous time moving averages driven by Lévy processes. We provide necessary and sufficient conditions on the kernel for the moving average to be a semimartingale in the natural filtration of the Lévy process, and when...

This paper presents a generalized pre-averaging approach for estimating the integrated volatility, in the presence of noise. This approach also provides consistent estimators of other powers of volatility -- in particular, it gives feasible ways to consistently estimate the asymptotic variance...

Strong approximation for sums of a class of stationary processes with optimal bound is established. The main tools are m-dependent approximation and block techniques. Some previous results are improved.

We consider a form of state-dependent drift condition for a general Markov chain, whereby the chain subsampled at some deterministic time satisfies a geometric Foster-Lyapunov condition. We present sufficient criteria for such a drift condition to exist, and use these to partially answer a...

We propose a discrete-time random walk on , d=1,2,..., as a variant of recent models of random walk on in a random environment which is i.i.d. in space-time. We allow space correlations of the environment and develop an analytic method to deal with them. We prove, under some general assumptions,...